D.C. Mayor Vincent C. Gray kicked off an ambitious effort Thursday to move 500 homeless families into permanent housing by summer, even as police struggled to learn more about two men who were found dead Tuesday at a makeshift camp just off Pennsylvania Avenue SE.

Relisha and her mother and siblings were among hundreds of families crowded into the shelter or others like it, or housed in motel rooms paid for by the city. Gray (D) has pledged to move 500 of the families into subsidized apartments by July 9, greatly expanding a program known as Rapid Rehousing that subsidizes apartments for families in four-month increments, often up to nearly a year.

Families pay 40 percent of their income for the housing; the city pays the rest.

To encourage landlords to participate and to allow families to stay for a year, the city says it will guarantee rental payments for 12-month leases; help evict tenants who don’t follow through on paying their share; and pay hundreds of dollars per apartment to bring dilapidated units up to code.

Law enforcement officials move a body found this week at the intersection of Pennsylvania Avenue and I-295 in Southeast. Authorities said hypothermia likely was a factor in the deaths. (Astrid Riecken/For The Washington Post)

The 100-day clock to house 500 families began running on April 1. Gray said landlords have offered nearly 200 apartments so far, but the units must be inspected, and some probably will be unusable. The city plan asks churches, synagogues and mosques to adopt one homeless family each, and to provide encouragement and support services to help people get back on their feet.

“I know that we can solve this,” Gray said. “We can’t stand by and watch the increasing prosperity of this city while at the same time we have a dismal experience for families who are living at the margins.”

Advocates for the homeless are split on the Rapid Rehousing initiative. The city has a record of keeping most families who participate from reappearing at shelters. But advocates say the program does not provide enough stability for families to focus on getting long-term jobs, degrees and other tools to keep from falling back into poverty.

The program is not designed to help homeless men living on the street, such as the two who were found near a highway overpass on Tuesday. The D.C. Medical Examiner’s Office has not yet said how they died, but police say it probably was hypothermia. Gray called the deaths tragic and said he was awaiting autopsies and more information. Police think they know the name of one of the men but are waiting for his relatives to identify him.

The mayor cautioned that hypothermia deaths are complicated and not always preventable. He said he accompanied emergency workers downtown on a frigid day in January in an effort to convince people living on the street to seek shelter. Police took some of the homeless men to hospitals against their will for mental health evaluations, Gray said.

“In the [homeless] singles population, we have a lot of people who have substance abuse problems, who have mental health problems, who have a combination of those things, and even under extreme conditions it can be very difficult to get them to come in out of the cold,” Gray said. “When you have those kinds of factors, it is very difficult to get people to make responsible decisions in their own best interest.”

Beverly Fields, a spokeswoman for the medical examiner, said there were five hypothermia deaths in 2012, 11 in 2013 and two confirmed this year. It was not clear how many of the 2013 deaths were from the most recent winter — Fields said her office might be able to provide a monthly breakdown, but not until at least Friday. Fields also cautioned that “every hypothermia death may not be a homeless person.”

The medical examiner includes hypothermia deaths in an annual report, usually released in July. At a hearing in February, the Department of Human Services said there is a one-year delay in receiving the report and verifying hypothermia-related deaths.

“We don’t often find out the final detail for several months, but that’s not a complaint,” DHS Director David Berns said. “What happens if you have a person who used a lot of drugs and passed out on the street — did you die from the drug overdose? Did you die on the street because you did not crawl under a blanket that was a few inches away? They have to look at all of the nuances of what is the proximate cause, what is the actual cause.”

Cornell Chappelle, deputy director of the Community Partnership for the Prevention of Homelessness, the contractor that runs District homeless shelters, said information about homeless deaths “has been really spotty — it takes a while.”

Fields said the medical examiner reaches out to homeless groups but that it’s incumbent on those groups to call the office for information.

“Our mission is cause and manner of death,” she said. “We don’t necessarily put reporting out on a daily basis.”

She said that when groups who work with the homeless contact the medical examiner, “we provide them with information and data.”

Berns and Chappelle said the bodies discovered Wednesday were found in an area that was not a known homeless area — and not an area that city contractors visited to look for people in need as temperatures dropped Tuesday night.

Berns said the city opened two overflow shelters for homeless singles on Tuesday night and placed 14 families in motel rooms.